Set Theory Problems: S1 U S2 = (S1' ∩ S2')' and S1 U S2 - (S1 ∩ S2') = S2

[sbc824]

Homework Statement

show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution

I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or S2, then x ∉ S1' and S2'
If x ∉ S1' and S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement

show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution

I have not attempted this as I'm not sure how to start this one...any help would be appreciated.

 

[SammyS]

Homework Statement

show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution

I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or x ∈ S2, then x ∉ S1' and x ∉ S2' This is not correct.

If x ∈ S1, then it does not need to be in S2. If it's not in S2, then x ∈ S2'.


If x ∉ S1' and x ∉ S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement

show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution

I have not attempted this as I'm not sure how to start this one...any help would be appreciated.

You don't have the first part right.

 

[sbc824]

You don't have the first part right.

wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.

 

[SammyS]

wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.

Another problem with your solution to part 1 is that you have only shown that S1 U S2 ⊆ (S1' ∩ S2')' (that is, if you have truly corrected your proof). To show equality, you also need to show that S1 U S2 ⊇ (S1' ∩ S2')' .